2.2. Permutation-Inversion Symmetry Operations

The relation between the geometric symmetry operations of
sect. 2.1 and the permutation-inversion
symmetry operations can be demonstrated most easily by considering an equation
relating the laboratory-fixed Cartesian coordinates
(XiYiZi) of the electrons and nuclei
in a diatomic molecule to the molecule-fixed electronic coordinates
(xyz), the equilibrium positions of the nuclei, the displacement vectors
di of the nuclei, and the two rotational angles
&theta"; φ
[15,16].

The left-hand sides of eqs (2.03) contain the Cartesian coordinates of an
electron and the two nuclei in an axis system parallel to that fixed in the
laboratory, but located at the center of mass of the equilibrium configuration
of the nuclei. The right-hand sides of eqs (2.3) all contain a
3 × 3 rotation matrix, the direction cosine matrix, which transforms
vector components from a molecule-fixed axis system to a laboratory-fixed axis
system. This matrix is a function of the rotational variables θ and
φ, specifying the direction of the
internuclear axis of the diatomic molecule in the laboratory-fixed axis system.
The column vector on the far right in the first of eqs (2.3) contains the
molecule-fixed coordinates of an electron. The second and third column vectors
on the far right contain the positions of the two nuclei in the molecule-fixed
axis system. At equilibrium
(d1 = d2 = 0)
both nuclei lie on the z axis, with the center of mass at the origin,
and with internuclear distance re;
µ = m1m2/(m1
+ m2) is the reduced mass of the molecule.

Consider now the effect of the four symmetry operations E,
συ(xz),
i, and C2(y) on the coordinates in
eqs (2.3). The transformations of the coordinates on
the right side of (2.3) can be obtained from
table 3 and
table 4 and from the text of
sect. 2.1. It is fairly easy to show
from (2.3) that these operations give rise to the transformations of
laboratory-fixed coordinates shown in table 5. From table 5 we see
that the geometric symmetry operation
συ(xz),
when it is applied to the electronic, vibrational. and rotational (i.e., to
all) variables, is equivalent to the laboratory-fixed inversion operation
I; and that the geometric symmetry operation
C2(y), when it is applied to the electronic,
vibrational, and rotational variables, is equivalent to the permutation
P of the two (identical) nuclei in the molecule. From table 5
we also see that the geometric symmetry operation i is equivalent to the
product P · I, i.e., to the combined permutation andlaboratory-fixed inversion operation.

TABLE 5. The effect of various symmetry operations on
laboratory-fixed Cartesian coordinates of the electrons and of the two
nuclei

Symmetryoperation

Coordinates acted upon

Xe

Ye

Ze

X1

Y1

Z1

X2

Y2

Z2

E

Xe

Ye

Ze

X1

Y1

Z1

X2

Y2

Z2

συ(xz)

- Xe

- Ye

- Ze

- X1

- Y1

- Z1

- X2

- Y2

- Z2

i

- Xe

- Ye

- Ze

- X2

- Y2

- Z2

- X1

- Y1

- Z1

C2(y)

+ Xe

+ Ye

+ Ze

+ X2

+ Y2

+ Z2

+ X1

+ Y1

+ Z1

It is evidently necessary to distinguish clearly between the "molecule-fixed"
inversion operation i, and the "laboratory-fixed" inversion operation I, since
these two operations are not equivalent. In particular, i exists only if
the diatomic molecule is homonuclear, whereas I exists for all diatomic
molecules. The precise difference between these two inversion operations can
only be understood after some study [14,15,16].

Rotational energy levels are said to be of even (+) parity if the corresponding
complete molecular wave functions are invariant to the laboratory-fixed
inversion operation I; they are of odd (-) parity if the wave functions
transform into their negatives. Rotational energy levels of homonuclear
diatomic molecules are said to be symmetric (s) if the corresponding
complete molecular wave functions are invariant to the exchange of identical
nuclei P; they are antisymmetric (a) if the wave functions
transform into their negatives.